I am just interested in knowing what should be the general mindset while solving/tackling a mathematical problem. I know that there is no one way of approaching the problem, but still. I was searching this online and found out about 'How to solve it' by Polya. Has anyone read the book? How is it? Can someone give any references or books that might help?
closed as too broad by user296602, Eric Wofsey, Surb, user299912, MathMajor Jul 2 '16 at 8:14
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Yes, there is. Mathematical problems encountered typically fall into certain categories such as arithmetic, trigonometry, quadratic equations, calculus etc. We learn to solve problems in each field by successful repetition of methods which are known to work.
When tackling a mathematical problem, the first step is to ask yourself what field of mathematics, which you have experience of, has something to contribute in the way of finding a solution. Typically students are only asked to solve problems for which they have already practiced the method.
Then, you must map the method which you have practiced, to the problem. This is the part where you often have to think outside the box because it may bot be obvious that some method is appropriate. Often, this will be asking yourself, "what exactly, in mathematical terms, is the question", or "what are the methods I have learnt, and how might they move me closer to an answer?" Look for the signatures of each method or type of mathematics, e.g. finding the maximum or minimum point of some continuous function will often be calculus. Sometimes a question is in words and you must first write out he facts in mathematical notation before it becomes obvious.
If you can't see a clear path to the end solution then it may be beneficial to apply what appears the most appropriate method and see if the solution becomes apparent. The more you practice, the more the solution will be obvious to you before you even start.
A picture tells a thousand words. If you can find a way to visualise your problem, such as a Venn diagram or to sketch a graph or plot the data, then this is almost always useful as it brings more powerful areas of the mind into play which can manipulate concepts, giving you a place to pin the relevant information and increasing your mind's capacity to remember while you work with it.
The final step in tackling a mathematical problem, is to verify your answer. This may be by solving the problem by a 2nd method, and ensuring you reach the same conclusion. Or it will often be by substituting your answer back into a part of the original problem, to ensure you get no result that contradicts the remainder of the question. The key in this final step is to identify a test which is very sensitive to any error.